Aniket, the uncertainty has an infinite number of meanings depending on applications. In Kalman filtering, uncertainrty is associate with unpredictable system or observation usually temporary errors which cannot be efficiently ignored by the linear Kalman algorithm. Therefore, robust estimators are required. The answer to your question is thus the following. The minimization of P leads to the optimal estimate that has nothing to do with uncertainties.

Hello Yuriy, u r right that min{P} leads to an optimal estimate. But my doubt was if the minimum variance estimate of x is orthogonal to the error in the estimate. Then it implies that the min. variance estimate of x is uncorrelated to its error. If this is the case, my question was how can a minimization of the P, will lead to an optimal estimate. Because P is the co-variance of the error in the estimate. I hope i am able to explain better what my doubt is

Aniket, it depends of what you think about P which represents the difference between the model and the estimate.

Generally P is not the error covariance matrix. It is the MSE matrix. If you minimize P, you go to the optimal Kalman filter.

Otherwise, if you think that the filter is unbiased, then P is the error covariance matrix. But, in fact, the difference has some bias. Thus P can be minimized to produce a minimum variance estimate that in fact is the same unique Kalman estimate.

X and Y are orthogonal if E(XY)=0. and X and Y are uncorelated

if cov(X,Y)=E(XY)-E(X).E(Y)=0. so i don't think Xand Y are orthogonal then they will be uncorelated. one more thing in real life problem how do you calculate the state error which is equal to X_true- X_predicted. we never knows the true value of state.

Chitta, that is right, we never know X_true! Moreover, we commonly do not know the noise statistics. A rival to the Kalman filter that ignores these characteristics is the unbiased FIR filter. This filter requires instead an optimal averaging interval that can be determined via measurement.

Hello Chitta, u r right that its not necessary for two orthogonal variables, X and Y, to be uncorrelated. But If the Expected value, E[x] or E[y], of either or both of the variables is zero, then the orthogonal variables will also be uncorrelated. Like in my problem the expected value of the error of the estimated variable, E[e], is zero. so the correlation between the two orthogonal variables, X_estimated and its Error is zero.

first, the error covariance "P" only gives the approximate initial idea of the error variance, it helps the system work correctly in acquisition process (usually P is much larger than Q ), the eventually the system will trasition to steady state where the accuracy only depends on Q and R. so actually, in the initialization of P, we have to make sure it is located in the right uncertainty, not too big or not too small, or it will lose the correct tracking of the variables.